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Mirrors > Home > MPE Home > Th. List > sucexeloni | Structured version Visualization version GIF version |
Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7751 does not require ax-un 7677. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.) |
Ref | Expression |
---|---|
sucexeloni | ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6332 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordsuci 7748 | . . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
4 | elex 3466 | . 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | |
5 | elong 6330 | . . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) | |
6 | 5 | biimparc 481 | . 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On) |
7 | 3, 4, 6 | syl2an 597 | 1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3448 Ord word 6321 Oncon0 6322 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-suc 6328 |
This theorem is referenced by: onsuc 7751 1on 8429 2on 8431 |
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